Two-dimensional mesh embedding for B-spline methods

Many factors motivate consideration of B-splines as basis functions for solving partial differential equations. These are arbitrary orders of accuracy and high resolving powers similar to those of compact schemes. Furthermore, if one uses a Galerkin scheme one gets, in addition to conservation of the discretized quantities, conservation of quadratic invariants such as energy. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. An algorithm is presented for constructing a global spline basis that automatically has d - 1 continuous derivatives at mesh-block boundaries as everywhere else (here d is the polynomial degree). The basis functions are simply suitable products of one-dimensional B-splines. Both integer and noninteger refinement ratios are allowed across mesh blocks. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented. (C) 1998 Academic Press.